According to Lemma REF , by virtue of hypothesis
(ii) \(x_0\in {\rm int}\, {\rm dom}\, \nu \)  and \(\nu \)  turns out to be bounded
from above in a neighbourhood of \(x_0\) .
According to Lemma REF , each function \(x\mapsto {\rm dist}\left(f(x,z),C\right)\) 
is convex on \(K\)  and therefore so is function \(\nu \) , which
can be regarded as an upper envelope of functions \({\rm dist}\left(f(\cdot ,z),C\right)\) 
over \(z\in K\) .
As a convex function, which is bounded from above in a neighbourhood
of \(x_0\) , \(\nu \)  must be continuous on the interior of its domain,
in the light of a well-known result in convex analysis (see, for
instance, [1]}).
